Referring to the discretization of partial differential equations using Finite Volume Method.

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59 views

Is the Finite Volume Method conservative when the source term depends on the variable?

I trying to do a simulation where there are two quantities, $\Delta\theta(x,t) = \theta(x,t) - \theta_{o}$ and $\Delta\nu(x,t) = \nu(x,t) - \nu_{o}$. These quantities are chemical concentrations ...
2
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2answers
113 views

Are the Finite Difference and Finite Volume Methods different after the application of the Gauss Divergence theorem on the FVM?

I have a rudimentary question about the differences between finite difference (FDM) and finite volume methods (FVM). In FDM we concentrate on the nodes (points) in space while in FVM we concentrate ...
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1answer
59 views

1D uniform flow test case for compressible flow

Are there any one dimensional hyperbolic test cases for a 1D uniform flow? (I need the test case for the purpose of showing the satisfaction of geometric conservation law on a moving mesh).
3
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1answer
102 views

a few questions on understanding geometric conservation law

I was wondering if anyone could help with understanding the geometric conservation law for moving domains. I came across Link1, and have tried to understood the paper by Farhat et al Link2. So far ...
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1answer
59 views

Order of accuracy of DGFEM or FEM

I know that it is possible to determine the theoretical order of accuracy (B) of numerical solutions in FVM (for instance for a steady problem in which only the central differencing scheme (CDS) is ...
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1answer
124 views

what is the difference between non-conformal and conformal?

So far what I understand is that two neighbouring elements are conformal if their edges and faces match exactly, whereas with non-conformal elements this is not the case. For instance, h-refinement ...
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1answer
72 views

Is it possible to show global conservative properties FEM as it is done in FVM?

I know that in FVM, it is possible to show that a discretisation scheme is conservative by adding the discrete terms over a few control volumes and showing that all terms cancel apart from those ...
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46 views

Do collocated grid arrangements definitely result in the checkerboard effect?

I understand the checkerboard effect due to the use of collocated grid arrangements in FVM. However, I wanted to know whether this problem is definitely bound to effect the results? For instance, I ...
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1answer
95 views

Are FEM or DGFEM methods based on integrals or PDEs?

I know that FVM is based on the integral form of conservation laws, and FDS is based on PDEs. What I'm confused by, is whether FEM and DGFEM formulations are based on integral or pde form of ...
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50 views

problem with understanding the fluid boundary conditions of a 1D probelm

I am having problems understanding the boundary conditions of the problem described in this paper on researchgate Essentially the problem consists of a one dimensional fluid chamber in contact with a ...
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0answers
50 views

difference/similarities between DGFEM and FVM and FEM methods

I have been to trying to understand the discontinous galerkin method. My understanding is that in comparison to FVM, the essential difference is in the weighting function used and by setting the ...
2
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1answer
37 views

Use of structs in Axisymmetric Finite Volume method

This might be better somewhere else, but I'll give it a try here first. I'm implementing a finite volume scheme for an axisymmetric problem in C, and am looking for a more efficient way to handle all ...
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1answer
78 views

Diffusion with space dependent drift in Fipy

I need to solve a diffusion equation in periodic boundary conditions using fipy but I would like to have a drift term that depends on the position so like this: $$ \partial_t u(x,t) = \partial_x^2 ...
2
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1answer
125 views

MAC Projection in Projection method?

My question concerns the following paper: A Conservative Adaptive Projection Method for the Variable Density Incompressible Navier–Stokes Equations ...
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40 views

Restarting the Integrator

I'm solving a diffusion-reaction PDE with discontinuous variable coefficient and source term (like a step function) at a point. I use FVM with harmonic average on the coefficient, but no special ...
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2answers
190 views

Dirichlet BCs - alternative implementation methods

I am having problems with solving a hyperbolic wave problem with Dirichlet BCs. I have tried reducing the time step sizes, which does not affect the results, and notices increasing the number of nodes ...
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0answers
45 views

should boundary conditions be effecting moving mesh results?

I have a question on the use of moving mesh to solve the inviscid euler equations. I have solved the following equations: $$\frac{\partial}{\partial t}\left[\begin{array}{c} \rho\\ \rho u ...
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1answer
80 views

How to define fluxes for two dimensional convection-diffusion equation?

I want to solve the following differential equation using control volume approach on a Cartesian mesh: $$\frac{\partial T}{\partial t} + \frac{\partial T}{\partial x} + \frac{\partial T}{\partial y}= ...
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1answer
100 views

FVM - virtual node discretisation

I have come across the paper titled: A monolithic fluid structure interaction algorithm ... For a 1D grid, at the boundaries the paper uses virtual nodes $x_{0}$ and $x_{N+1}$ (page 372) and for ...
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75 views

Discontinuity at Interface

The equation at the left of the interface is \begin{equation} \displaystyle\frac{\partial C_i}{\partial t} = D_i \nabla^2 C_i - z_i \frac{D_i}{RT}F \nabla \cdot (C_i \nabla \phi_2) \end{equation} ...
3
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1answer
62 views

linear stability analysis using spectral radius

I am analysing the stability of a series of 1D linear equations of the form \begin{equation} \frac{d}{dt} x = A x \end{equation} discretised using upwind and central finite volume methods, etc, with ...
4
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1answer
96 views

Flux at coarse-fine mesh grid interface?

I am trying to solve one dimensional inviscid Burger's equation using adaptive mesh refinement. This is the PDE: $$\frac{\delta U}{\delta t} + \frac{\delta F}{\delta x} = 0$$ where the flux F of the ...
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76 views

Is it generally unstable to use in multidimensional simulations finite difference schemes with higher orders than 2?

I'm part of a team trying to generalize a 1D Advection-Diffussion-Reaction code we inherited by extending it to 2D by using dimensional splitting, i.e. solving advection and diffusion for x and y ...
4
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1answer
83 views

transverse component for multidimensional advection in method of lines

So I inherited from some people a code that solves the advection-diffusion-reaction equation for a particular system. The original code was first implemented in 1D which worked fine in cartesian ...
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2answers
358 views

The meaning of conservative discretization in Galerkin FEM and Discontinuous Galerkin

I do understand the meanning of "conservative discretization" within the FVM/FDM framework, indeed it is well explained in this post. Now, according to the table in this slide (pp.4), it concludes: ...
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53 views

Estimating the local compression/expansion ratio for a transformation on a point cloud

Let's say we have an unorganized point cloud P1 with N points, each with coordinates {x,y,z}. We apply non-rigid transformation to P1 (translation + rotation + warping), to obtain point cloud P2. ...
2
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1answer
195 views

Add User-defined/custom differential equations in OpenFoam (CFD)

I am new to OpenFoam. And I am trying to add a set (user defined) of differential equations to OpenFoam. I want to solve this user defined set of equations at each time point in addition to standard ...
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163 views

What is the difference between SSPRK3 and RK3 time discretization methods?

Here, SSPRK3 refers to third order strong stability preserving Runge-Kutta and RK3 refres to regular third order Runge-Kutta method. The meaning of the method is obvious from the name. However there ...
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1answer
141 views

Temperature dependent 1-d conduction in Python?

I'm trying to write a Python code that is a numerical solver for 1-d heat conduction (using FVM) with a temperature dependent thermal conductivity. The solver has three functions I need to iterate ...
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1answer
109 views

What does “strongly conservative” mean in the context of numerical methods?

I have a homework problem that asks me to show that 1st order unwinding or central differencing can give a strongly conservative, consistent scheme for the 1-D Burger's Equation using a finite volume ...
4
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2answers
354 views

Manufactured solution for pressure based 3d incompressible Navier-Stokes solver with wall boundaries

I already successfully verified my solver (SIMPLE-type FVM-method) with the following manufactured solution (3d Taylor-Green vortex) on the solution domain $[-1,1]^3$ with Dirichlet boundary ...
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0answers
102 views

Arbitrary Choosing of the Solution Domain - Navier Stokes and Manufactured Solutions

I want to verify a finite-volume solver (SIMPLE-Algorithm) for the incompressible Navier-Stokes equations by using a manufactured solution. I use Dirichlet boundary conditions for the velocity at all ...
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33 views

How to implement two-point viscous fluxes in FVM context

I am trying to implement the two-point viscous fluxes in an in-house unstructured code (FV, all hex cells). It will be really helpful if anyone can explain the algorithm for non-orthogonal meshes. ...
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1answer
61 views

Application of CLAWPACK to Richards' equation

I'm looking to solve the Richards' equation. This models water flow in porous media and is a nonlinear, possibly degenerative, parabolic differential equation that takes the form $\partial_t ...
2
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1answer
106 views

Unable to validate the Roe matrix for the Shallow Water Equations

In LeVeque's Finite Volume Methods for Hyperbolic Problems, p. 320-321, one may find the derivation of the Roe matrix to the 1D Shallow Water Equations (SWEs). It is $$ ...
2
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0answers
155 views

Finite Volume/difference scheme for 2d continuity equation

I am trying to solve the following 2D continuity equation on the rectangular domain [0,1]x[0,1] using the finite volume method. $\rho_t + \nabla \cdot (\rho v) = 0$ where the velocity $v =(v1,v2)$ is ...
2
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1answer
162 views

Finite volume method implementation issues

I am trying to write a simple finite volume method code but there are some concepts I'm still not really getting right (perhaps I'm overcomplicating things) Given a uniform grid, the idea is to ...
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1answer
81 views

Finite Volume Method flux integration

I was reading through this document about FVM. I understood all up to the point where we have on page 15 the following $$(\bar u^{n+1}_i - \bar u^n_i) \Delta x + \int^{t^{n+1}}_{t^n} ...
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1answer
189 views

What kind of test cases are convenient to use for testing the code for Euler equations of gas dynamics in polar coordinates?

Consider Euler equation of gas dynamics in polar coordinates as $$ \left( \begin{array}{ccc} \rho \\ \rho u_{r} \\ \rho u_{\theta} \\ E \end{array} \right)_{t} + \frac{1}{r} \left( ...
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119 views

Euler Equation Eigensystem with Gravity in the Energy Flux

I am modifying a conservative form of the Euler equations with gravity in the energy flux (see previous question: Energy Conservation in Conservation Laws with Source Terms) for use in a Riemann ...
2
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1answer
222 views

Energy Conservation in Conservation Laws with Source Terms

I'm wondering if anyone can help me understand energy conservation when using conservation law methods (i.e. Riemann solver, High-Resolution Wave-Propagation Methods) with the addition of source ...
3
votes
1answer
407 views

Numerically solve a PDE in Python with a term calculated by coarse-graining

I'm trying to solve a PDE in Python of the form, $\dfrac{\partial c(\mathbf{x}, t)}{\partial t} = \mathrm{D} \nabla^2 c(\mathbf{x}, t) -\gamma \rho(\mathbf{x}, t) c(\mathbf{x}, t)$ where $c$ ...
0
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1answer
415 views

a circular plot from a vector which represents the temperature along the radius surface, which is the same for every radius

I have calculated the temperature of the section of a cylinder, which is subjected to a heat flow on its upper surface. Getting the temperature distribution in the 2D section. As shown in the ...
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1answer
57 views

Accurate dot-product of fields with only knowing normals

I am trying to accurately calculate $\vec{j} \cdot \vec{E}$ for an electron energy equation on a finite-volume mesh. $\vec{j}$ is the electron current density, and $\vec{E}$ is the electric field. ...
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2answers
127 views

Is there a bound on the number of edges, facets, and elements in a 3D simplicial mesh in terms of the number of mesh nodes?

I am currently working on a 2D finite element code with a mesh that contains duplicate nodes at certain interfaces where jumps occur. In order to set up the appropriate linear systems, I have to take ...
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2answers
139 views

Finite element: handle discontinuity/abrupt interface

I am using Galerkin's method to set up some pde solver for my problems(1D/2D) at hand. Some abrupt material interfaces do exist, so far, what I did regarding this is simply: 1) no element across the ...
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114 views

boundary conditions with non-constant coefficients in cell centred finite volume method

Suppose am solving the heat conduction equation in 1d with Dirichlet boundary conditions. The thermal conductivity $k$ is a non constant function. So $-(k(x)u'(x))' = f(x)$ The value of $k$ enters ...
3
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1answer
675 views

Comparison of Lattice Boltzmann Method vs Traditional Navier-Stokes based Methods

I have a choice of two options, analysing and implementing Lattice Boltzmann methods or traditional Navier Stokes based methods. I'm a CFD newbie and I have a rough idea (though not rigorous enough to ...
2
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1answer
262 views

Moving airfoil boundary conditions

I am trying to simulate a moving airfoil with constant speed (Mach=0.755, aoa=1.25). I solve Euler equations with Roe's method. I have two boundary conditions: Farfield Slip wall (airfoil) For all ...
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194 views

finite volume for diffusion equation with anisotropic (tensor) coefficient

Consider the scalar PDE for $u$ with Dirichlet boundary conditions: $\mathrm{div}(\mathcal{K}\nabla u) = f\; \forall x\; \in \Omega \subset R^2$, $u = 0 \; \forall \; x\;\in \partial\Omega$ ...